Stability analysis and control of discrete-time systems with delay

The research presented in this thesis considers the stability analysis and control of discrete-time systems with delay. The interest in this class of systems has been motivated traditionally by sampled-data systems in which a process is sampled periodically and then controlled via a computer. This setting leads to relatively cheap control solutions, but requires the discretization of signals which typically introduces time delays. Therefore, controller design for sampled-data systems is often based on a model consisting of a discrete-time system with delay. More recently the interest in discrete-time systems with delay has been motivated by networked control systems in which the connection between the process and the controller is made through a shared communication network. This communication network increases the flexibility of the control architecture but also introduces effects such as packet dropouts, uncertain time-varying delays and timing jitter. To take those effects into account, typically a discrete-time system with delay is formulated that represents the process together with the communication network, this model is then used for controller design While most researchers that work on sampled-data and networked control systems make use of discrete-time systems with delay as a modeling class, they merely use these models as a tool to analyse the properties of their original control problem. Unfortunately, a relatively small amount of research on discrete-time systems with delay addresses fundamental questions such as: What trade-off between computational complexity and conceptual generality or potential control performance is provided by the different stability analysis methods that underlie existing results? Are there other stability analysis methods possible that provide a better trade-off between these properties? In this thesis we try to address these and other related questions. Motivated by the fact that almost every system in practice is subject to constraints and Lyapunov theory is one of the few methods that can be easily adapted to deal with constraints, all results in this thesis are based on Lyapunov theory. In Chapter 2 we introduce delay difference inclusions (DDIs) as a modeling class for systems with delay and discuss their generality and advantages. Furthermore, the two standard stability analysis results for DDIs that make use of Lyapunov theory, i.e., the Krasovskii and Razumikhin approaches, are considered. The Krasovskii approach provides necessary and sufficient conditions for stability while the Razumikhin approach provides conditions that are relatively simple to verify but conservative. An important conclusion is that the Razumikhin approach makes use of conditions that involve the system state only while those corresponding to the Krasovskii approach involve trajectory segments. Therefore, only the Razumikhin approach yields information about DDI trajectories directly, such that the corresponding computations can be executed in the low-dimensional state space of the DDI dynamics. Hence, we focus on the Razumikhin approach in the remainder of the thesis. In Chapter 3 it is shown that by considering each delayed state as a subsystem, the behavior of a DDI can be described by an interconnected system. Thus, the Razumikhin approach is found to be an exact application of the small-gain theorem, which provides an explanation for the conservatism that is typically associated with this approach. Then, inspired by the relation of DDIs to interconnected systems, we propose a new Razumikhin-type stability analysis method that makes use of a stability analysis result for interconnected systems with dissipative subsystems. The proposed method is shown to provide a trade-off between the conceptual generality of the Krasovskii approach and the computationally convenience of the Razumikhin approach. Unfortunately, these novel Razumikhin-type stability analysis conditions still remain conservative. Therefore, in Chapter 4 we propose a relaxation of the Razumikhin approach that provides necessary and sufficient conditions for stability. Thus, we obtain a Razumikhin-type result that makes use of conditions that involve the system state only and are non-conservative. Interestingly, we prove that for positive linear systems these conditions equivalent to the standard Razumikhin approach and hence both are necessary and sufficient for stability. This establishes the dominance of the standard Razumikhin approach over the Krasovskii approach for positive linear discrete-time systems with delay. Next, in Chapter 5 the stability analysis of constrained DDIs is considered. To this end, we study the construction of invariant sets. In this context the Krasovskii approach leads to algorithms that are not computationally tractable while the Razumikhin approach is, due to its conservatism, not always able to provide a suitable invariant set. Based on the non-conservative Razumikhin-type conditions that were proposed in Chapter 4, a novel invariance notion is proposed. This notion, called the invariant family of sets, preserves the conceptual generality of the Krasovskii approach while, at the same time, it has a computational complexity comparable to the Razumikhin approach. The properties of invariant families of sets are analyzed and synthesis methods are presented. Then, in Chapter 6 the stabilization of constrained linear DDIs is considered. In particular, we propose two advanced control schemes that make use of online optimization. The first scheme is designed specifically to handle constraints in a non-conservative way and is based on the Razumikhin approach. The second control scheme reduces the computational complexity that is typically associated with the stabilization of constrained DDIs and is based on a set of necessary and sufficient Razumikhin-type conditions for stability. In Chapter 7 interconnected systems with delay are considered. In particular, the standard stability analysis results based on the Krasovskii as well as the Razumikhin approach are extended to interconnected systems with delay using small-gain arguments. This leads, among others, to the insight that delays on the channels that connect the various subsystems can not cause the instability of the overall interconnected system with delay if a small-gain condition holds. This result stands in sharp contrast with the typical destabilizing effect that time delays have. The aforementioned results are used to analyse the stability of a classical power systems example where the power plants are controlled only locally via a communication network, which gives rise to local delays in the power plants. A reflection on the work that has been presented in this thesis and a set of conclusions and recommendations for future work are presented in Chapter 8

Invariance positive et observateur intervalles appliqués aux systèmes linéaires à retards sous contraintes

RÉSUMÉ Cette thèse porte sur l’étude du problème de la commande avec contraintes des systèmes linéaires continus à retards. Deux approches dans la littérature se sont données à développer des méthodes adéquates pour examiner la stabilité et contribuer à des procédures et outils de stabilisation. La première, considère l’effet de la saturation, tandis que la deuxième approche, basée sur la théorie d’invariance positive, repose principalement sur la conception d’une loi de contrôle non saturante ayant un comportement linéaire dans le domaine des contraintes. Des résultats concernant l’application du concept d’invariance positive à la stabilisation des systèmes à retards soumis à des contraintes ont été développés, mais restent restrictifs, de fait qu’ils sont indépendants du retard, un paramètre essentiel du système. On développe alors, dans la première partie de cette thèse des conditions nécessaires et suffisantes dépendantes du retard afin de garantir l’invariance positive de domaine des contraintes par rapport aux trajectoires de systèmes autonomes à retard. Ce résultat repose sur la transformation de premier ordre, basée sur la formule de Newton-Leibniz, de système original à retard discret, en un système à retard distribué. Une fonction de Lyapunov-Razumikhin associée au système à retard distribué garantissant la stabilité asymptotique dépendante de retard de système original est proposée. L’objectif principal visé dans la deuxième partie de cette thèse est d’appliquer le résultat du concept d’invariance positive dépendante du retard au problème de la commande sous contraintes, dissymétriques ainsi que symétriques, des systèmes à retards. Ainsi des conditions permettant la synthèse d’un régulateur par retour d’état stabilisant le système en boucle fermée en présence des contraintes, sont données. Ces conditions permettent de formuler un algorithme basé sur des schémas de Programmation Non Linéaire (NLP), ayant pour objectif la détermination du régulateur stabilisant le système en boucle fermée avec une borne maximale du retard. En effet la loi de retour d’état calculée assure, d’une part, la stabilité asymptotique de système sans retard, et d’autre part, la maintenir pour une valeur d’une borne maximale de retard, tout en respectant les contraintes : c’est la loi de commande sous contraint robuste vis à vis le retard. Les résultats obtenus sont intéressants et plus généraux que ceux développés dans la littérature. La troisième partie de cette thèse montre, pour la première fois à notre connaissance, que les observateurs intervalles, en appliquant le concept d’invariance positive, peuvent apporter des réponses intéressantes au problème de la commande sous contraintes des systèmes linéaires à retards, variable dans le temps.----------ABSTRACT In this thesis, the stabilization problem of linear continuous-time delay system with constrainted control is studied. There are two main approaches in the literature dealing with the problem of performance and stability of dynamical constrained control systems. The first one considers the effect of saturation while guaranteeing asymptotic stability. The second one, so-called positive invariance approach, is based on the design of the control law which works inside a region of linear behavior where saturations do not occur. Most of the works related to positive invariance concept have been developed for time delay systems with constrained control, but remain so restrictive, given that they are independent of delay, which is an essential parameter of the system. In the first part of this thesis, the necessary and sufficient algebraic conditions with delay dependence allowing to obtain the largest positively invariant set of delay system are given. The results can include information on the size of delay, and therefore, can be delay dependence positively invariant conditions. Based on the Newton-Leibniz formula, these results use a transformation form an original system with discrete delay to a system with distributed delay. A Lyapunov-Razumikhin function for system with distributed delay, in order to guarantee the asymptotic stability of the original system is proposed. The second part of this thesis, is to apply the concept of the delay dependent positive invariance to the robust regulator problem of continuous time delay system with symmetric and non-symmetric constraints. In fact the synthesis of state-feedback controllers is solved based on delay-dependent positively invariant set of system in closed-loop. We first obtain the necessary and sufficient algebraic conditions with delay dependence allowing to obtain the largest positively invariant set of delay systems, then we convert the constrained control problem into a Non-Linear Programming (NLP) problem with delay the objective function to be maximized. Indeed the control is firstly chosen in order to stabilize the closed loop system, free of delay, then to guarantee the asymptotic stability of the closed loop system with delay-dependence. To the best of our knowledge, it is the first time, that the output stabilization problem for time-varying delay systems with constrained control based on the interval observer technique by using the dependent delay positive invariance concept is studied. Hence, first both matrices observer gain, the lower and the upper, are obtained by solving a Sylvester’s matrix equation. Second, the interval observer is developed and guaranteed the positivity of the upper and lower observations errors

Stability analysis and stabilization of linear aperiodic sampled-data systems subject to input constraints

Motivados pelo crescente uso de controladores embarcados em diferentes aplicações, onde um protocolo de comunicação é responsável pela transmissão de dados entre algoritmos computacionais, atuadores e sensores, a análise e o controle de sistemas amostrados foram abordados em muitos trabalhos. Nesse contexto, a amostragem aperiódica pode ser vista como uma abstração matemática empregada para representar, na teoria, o efeito de imperfeições no canal de comunicação, como instabilidades, flutuações e, em alguns casos, perda de pacotes de dados. Além disso, devido a limitações físicas dos atuadores, restrições de entrada e, em particular, a saturação são onipresentes em problemas reais de controle. Essas restrições são fonte de comportamentos não-lineares e de degradação do desempenho. Em muitos casos, apenas a estabilidade local (ou regional) do sistema em malha fechada pode ser garantida na presença de restrições e não-linearidades de entrada, mesmo para plantas lineares. Este trabalho lida com sistemas lineares amostrados aperiodicamente em que a entrada de controle, sujeita a restrições (por exemplo, saturação), é calculada com base em uma realimentação de estados do sistema. Dois problemas principais são abordados. O primeiro consiste na análise de estabilidade da origem de tais sistemas com a determinação de estimativas da região de atração da origem (RAO). O segundo, por sua vez, corresponde ao projeto de controle, onde uma lei de controle de realimentação de estados é calculada para otimizar o tamanho de uma estimativa da RAO do sistema em malha fechada resultante. Os métodos propostos são baseados no uso de programação semidefinida ou linear e, portanto, podem ser facilmente aplicados na prática. Um dos métodos propostos considera uma realimentação de estados linear sujeita a saturação e funções de Lyapunov quadráticas, resultando em estimativas elipsoidais da RAO do sistema. Dois outros métodos lidam com a análise de estabilidade do sistema amostrado sujeito a saturação fornecendo estimativas poliedrais da RAO. Devido à sua flexibilidade, a adoção de poliedros em vez de elipsóides permite uma redução de conservadorismo, mas é muito exigente em termos de complexidade computacional. Motivada por esse fato, esta tese também propõe um método de projeto de controle baseado em uma estratégia alternativa, onde a complexidade dos poliedros é fixada a priori. Essa ideia resulta em um problema de otimização com restrições bilineares, onde uma lei de controle linear por partes estabilizadora de complexidade relativamente baixa é encontrada para o sistema amostrado. Os métodos mencionados acima consideram uma abordagem não-estocástica, onde limites inferior e superior são impostos para o intervalo de amostragem do sistema, o qual é desconhecido e variante no tempo. Como contribuição adicional, esta tese também considera uma abordagem estocástica. Um método de projeto de controle é proposto para a estabilização global no sentido quadrático médio do sistema amostrado, onde a lei de realimentação de estados linear é sujeita a não-linearidades que satisfazem a uma condição de setor e os intervalos de amostragem correspondem a variáveis aleatórias com a distribuição de Erlang. A possibilidade de perda de pacotes de dados também é explicitamente levada em consideração através da distribuição de Bernoulli. Além disso, o método proposto, que se baseia na teoria de processos de Markov determinísticos por partes, resulta em condições de estabilização não-conservadoras no caso linear sem restrições de entrada.Motivated by the growing use of embedded controllers in different applications, where a communication protocol is responsible for the transmission of data between computer algorithms, actuators and sensors, the analysis and control design for sampled-data control systems have been addressed in many works. In this context, aperiodic sampling can be seen as a modeling abstraction employed to represent, in a theoretical framework, the effect of imperfections on the communication channel such as sampling jitters, fluctuations and, in some cases, packet dropouts. Moreover, due to physical limitations of actuators, input constraints and, in particular, input saturation are ubiquitous in real control problems. These constraints are source of nonlinear behaviors and performance degradation. In many cases, only local (or regional) stability of the closed-loop system can be ensured in the presence of actuators constraints and nonlinearities, even for linear plants. This work deals with linear aperiodic sampled-data systems where the control input, subject to constraints (e.g. saturation), is computed based on a feedback of the system state. It focuses on two main problems. The first one regards the stability analysis of the origin of such systems, with the determination of estimates of the region of attraction of the origin (RAO). The second one, in turn, corresponds to the control design, where a state-feedback control law is computed in order to enlarge an estimate of the RAO of the resulting closed-loop system. The proposed methods are based on the use of semidefinite or linear programming and can therefore be easily applied in practice. One of the proposed methods considers a linear saturating feedback of the system state and quadratic Lyapunov functions, leading to ellipsoidal estimates of the RAO of the system. Two other methods deal with the stability analysis of the sampled-data system subject to input saturation providing polyhedral estimates of the RAO. Because of their flexibility, adopting polyhedrons instead of ellipsoids allows a reduction of conservatism, but is very demanding in terms of computational complexity. Motivated by this fact, this thesis also proposes a control design method based on an alternative strategy, where the complexity of the polytopes is fixed a priori. This idea results in an optimization problem with bilinear constraints, where a stabilizing piecewise linear control law of relatively low complexity is found for the sampled-data system. The aforementioned methods consider a non-stochastic framework, where lower and upper bounds are imposed for the unknown, time-varying sampling interval of the system. As an additional contribution, this thesis also considers a stochastic setting. A control design method is proposed for the global stabilization in the mean square sense of the sampled-data system, where the linear feedback control law is subject to sector bounded nonlinearities and the sampling intervals are assumed to be random variables with the Erlang distribution. The possibility of packet dropouts is also explicitly taken into account through the Bernoulli distribution. Moreover, the proposed approach, which is based onthe framework of Piecewise Deterministic Markov Processes, leads to non-conservative stabilization conditions in the unconstrained linear case.Motivé par l’utilisation croissante de contrôleurs embarqués dans différentes applications, où un protocole de communication est responsable par la transmission de données entre les algorithmes numériques, les actionneurs et les capteurs, l’analyse et la conception de contrôle pour les systèmes de contrôle échantillonnées ont été abordées dans de nombreux travaux. Dans ce contexte, l’échantillonnage apériodique peut être considéré comme une abstraction mathématique employée pour représenter, dans un cadre théorique, l’effet des imperfections sur le canal de communication telles que la gigue d’échantillonnage, les fluctuations et, dans certains cas, les pertes de paquets. De plus, en raison des limitations physiques des actionneurs, les contraintes d’entrée et, en particulier, la saturation des entrées sont omniprésentes dans les problèmes de contrôle réels. Ces contraintes sont une source de comportements non-linéaires et de dégradation de la performance. Dans de nombreux cas, seule la stabilité locale (ou régionale) du système en boucle fermée peut être assurée en présence de contraintes et de non-linéarités des actionneurs, même pour les systèmes linéaires. Ce travail traite des systèmes linéaires échantillonnées apériodiquement où l’entrée de commande, soumise à des contraintes (par exemple la saturation), est calculée sur la base d’un retour d’état du système. Il se concentre sur deux problèmes principaux. Le premier consiste en l’analyse de stabilité de l’origine de tels systèmes avec la détermination d’estimations de la région d’attraction de l’origine (RAO). Le deuxième, à son tour, correspond à la conception de la commande, où une loi de commande à retour d’état est calculée afin d’agrandir une estimation de la RAO du système en boucle fermée résultant. Les méthodes proposées sont basées sur la programmation semi-définie ou linéaire et peuvent donc être facilement appliquées dans la pratique. L’une des méthodes proposées considère un retour d’état linéaire soumis à la saturation et des fonctions de Lyapunov quadratiques, conduisant à des estimations ellipsoïdales de la RAO du système. Deux autres méthodes traitent de l’analyse de stabilité du système échantillonné soumis à la saturation des entrées fournissant des estimations polyédriques de la RAO. En raison de leur flexibilité, l’adoption de polyèdres au lieu d’ellipsoïdes permet une réduction du conservatisme mais est très exigeante en termes de complexité de calcul. Motivée par ce fait, cette thèse propose également une méthode de conception de contrôle basée sur une stratégie alternative, où la complexité des polyèdres est fixée a priori. Cette idée se traduit par un problème d’optimisation avec des contraintes bilinéaires, où une loi de commande linéaire par morceaux stabilisante de complexité relativement faible est trouvée pour le système échantillonné. Les méthodes mentionnées ci-dessus considèrent un cadre non stochastique, où des limites inférieure et supérieure sont imposées pour l’intervalle d’échantillonnage inconnu et variable dans le temps du système. Comme contribution supplémentaire, cette thèseconsidère également un cadre stochastique. Une méthode de conception de contrôle est proposée pour la stabilisation globale dans le sens quadratique moyen du système échantillonné, où la loi de contrôle linéaire de retour d’état est soumise à des non-linéarités délimitées par secteur et les intervalles d’échantillonnage sont supposés être des variables aléatoires avec la distribution d’Erlang. La possibilité de pertes de paquets est aussi explicitement prise en compte via la distribution de Bernoulli. De plus, l’approche proposée, qui est basée sur le cadre des processus de Markov déterministes par morceaux, conduit à des conditions de stabilisation non conservatrices dans le cas linéaire sans contraintes

Model predictive control of grid-connected voltage source converters

In this thesis, the main focus is on the design and implementation of an advanced control scheme, namely model predictive control (MPC) to the grid- connected voltage source converter (VSC) for a three phase system. MPC is a control paradigm that solves a mathematical optimization problem based on a dynamic model of the system. Due to the computationally demanding nature of MPC, the areas of applications have long been restricted to slow dynamical systems. However, with the recent advancement of microprocessor and simu- lation technologies, application of MPC is now even possible for the control of power electronics. With a very powerful concept such as on-line cost optimisation, input/output constraint handling and model-based design, MPC is able to offer the optimal actuation that allows one to achieve very fast dynamics, while also considering uncertainties such as system parameter variations and unknown disturbances. Furthermore, it is also possible to take advantage of the discrete nature of the power converters and choose from the possible switching states the optimal solution according to the minimization of a predefined cost.  Exploring these advantages of MPC and making them suitable for the control of power converters are the key focus of the thesis. The first part of the thesis investigates a multi-variable control scheme, namely a predictive voltage controller that controls both DC bus voltage and re- active current (i.e. q-axis current) in the synchronous reference frame. Explicit tuning methods of MPC are introduced to improve the closed-loop transient response as well as improving the robustness against the parameter variations such as the grid inductance. The second part of the thesis focuses on the predictive current control design. A predictive current controller for VSC with LCL (inductor-capacitor- inductor) input filter is first proposed with a robust control scheme that employs nominal and disturbance rejection control parts. The nominal control part is designed using the reduced-order model (i.e. L filter model) to control dominant dynamics of the LCL filter where as the disturbance rejection control part actively suppresses the disturbance due to unmodeled dynamics of LCL filter (i.e. resonance of the LCL filter). Following from this, a predictive resonant controller is presented to control the converter in the stationary frame axis. A resonant module with a grid frequency is embedded in the model to handle the periodicity in the measured states and the reference inputs. The proposed de- sign considers the periodic input constraints in the stationary frame as well as disturbances due to grid voltage distortion. The last part of the thesis investigates the stability aspect of a finite control set predictive control (FCS-MPC) method and presents a design framework to handle the imposed the output current constraints in the cost function. All of the presented control methods in this thesis are experimentally validated on a 1kW prototype converter that has been built by the author

Contributions to nonlinear system modelling and controller synthesis via convex structures

Esta tesis discute diferentes metodologías de modelado para extraer mejores prestaciones o resultados de estabilidad que aquéllas que el modelado convencional basado en sector no-lineal de sistemas Takagi-Sugeno (también denominados cuasi-LPV) es capaz de producir. En efecto, incluso si las LMIs pueden probar distintas cotas de prestaciones o márgenes de estabilidad (tasa de decaimiento, $\mathcal H_\infty$, etc.) para sistemas politópicos, es bien conocido que las prestaciones probadas dependen del modelo elegido y, dado un sistema no-lineal, dicho modelo politópico no es único. Por tanto, se presentan exploraciones hacia cómo obtener el modelo que es menos perjudicial para la medida de prestaciones elegida. Como una última contribución, mejores resultados son obtenidos mediante la extensión del modelado politópico Takagi-Sugeno a un marco de inclusiones en diferencias cuasi-convexas con planificación de ganancia. En efecto, una versión sin planificación de ganancia fue propuesta por un equipo de investigadores de la Universidad de Sevilla (Fiaccini, Álamo, Camacho) para generalizar el modelado politópico, y esta tesis propone una version aún más general de algunos de dichos resultados que incorpora planificación de ganancia.This thesis discusses different modelling methodologies to eke out best performance/stability results than conventional sector-nonlinearity Takagi-Sugeno (also known as quasi-LPV) systems modelling techniques are able to yield. Indeed, even if LMIs can prove various performance and stability bounds (decay rate, $\mathcal H_\infty$, etc.) for polytopic systems, it is well known that the proven performance depends on the chosen model and, given a nonlinear dynamic systems, the polytopic embeddings available for it are not unique. Thus, explorations on how to obtain the model which is less deletereous for performance are presented. As a last contribution, extending the polytopic Takagi-Sugeno setup to a gain-scheduled quasi-convex difference inclusion framework allows to improve the results over the polytopic models. Indeed, the non-scheduled convex difference inclusion framework was proposed by a research team in University of Seville (Fiacchini, Alamo, Camacho) as a generalised modelling methodology which included the polytopic one; this thesis poses a further generalised gain-scheduled version of some of these results.Aquesta tesi discuteix diferents metodologies de modelatge per extreure millors prestacions o resultats d'estabilitat que aquelles que el modelatge convencional basat en sector no-lineal de sistemes Takagi-Sugeno (també anomenats quasi-LPV) és capaç de produir. En efecte, fins i tot si les LMIs poden provar diferents cotes de prestacions o marges d'estabilitat (taxa de decaïment, $\mathcal H_\infty$, etc.) per a sistemes politòpics, és ben conegut que les prestacions provades depenen del model triat i, donat un sistema no-lineal, el dit model politòpic no és únic. Per tant, es presenten exploracions cap a com obtenir el model que és menys perjudicial per a la mesura de prestacions triada. Com una darrera contribució, millors resultats són obtinguts mitjançant l'extensió del modelatge politòpic Takagi-Sugeno a un marc d'inclusions en diferències quasi-convexes amb planificació de guany. En efecte, una versió sense planificació de guany va ser proposada per un equip d'investigadors de la Universitat de Sevilla (Fiaccini, Álamo, Camacho) per a generalitzar el modelatge politòpic, i aquesta tesi proposa una versió més general d'alguns d'aquests resultats que incorpora planificació de guany.Robles Ruiz, R. (2018). Contributions to nonlinear system modelling and controller synthesis via convex structures [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/100848TESI

Unconstrained and constrained stabilization of bilinear discrete-time systems using polyhedral Lyapunov functions

The constrained and unconstrained stabilisation problem of discrete-time bilinear systems is investigated. Using polyhedral Lyapunov functions, conditions for a polyhedral set to be both positively invariant and domain of attraction for systems with second-order polynomial nonlinearities are first established. Then, systematic methods for the determination of stabilising linear feedback for both constrained and unconstrained bilinear systems are presented. Attention is drawn to the case where no linear control law rendering the pre-specified desired domain of attraction positively invariant exists. For this case, an approach guaranteeing the existence of a possibly suboptimal solution is established

Unconstrained and constrained stabilization of bilinear discrete-time systems using polyhedral Lyapunov functions

The constrained and unconstrained stabilisation problem of discrete-time bilinear systems is investigated. Using polyhedral Lyapunov functions, conditions for a polyhedral set to be both positively invariant and domain of attraction for systems with second-order polynomial nonlinearities are first established. Then, systematic methods for the determination of stabilising linear feedback for both constrained and unconstrained bilinear systems are presented. Attention is drawn to the case where no linear control law rendering the pre-specified desired domain of attraction positively invariant exists. For this case, an approach guaranteeing the existence of a possibly suboptimal solution is established

Assessing plant design with regards to MPC performance using a novel multi-model prediction method

Model Predictive Control (MPC) is nowadays ubiquitous in the chemical industry and offers significant advantages over standard feedback controllers. Notwithstanding, projects of new plants are still being carried out without assessing how key design decisions, e.g., selection of production route, plant layout and equipment, will affect future MPC performance. The problem addressed in this Thesis is comparing the economic benefits available for different flowsheets through the use of MPC, and thus determining if certain design choices favour or hinder expected profitability. The Economic MPC Optimisation (EMOP) index is presented to measure how disturbances and restrictions affect the MPC’s ability to deliver better control and optimisation. To the author’s knowledge, the EMOP index is the first integrated design and control methodology to address the problem of zone constrained MPC with economic optimisation capabilities (today's standard in the chemical industry). This approach assumes the availability of a set of linear state-space models valid within the desired control zone, which is defined by the upper and lower bounds of each controlled and manipulated variable. Process economics provides the basis for the analysis. The index needs to be minimised in order to find the most profitable steady state within the zone constraints towards which the MPC is expected to direct the process. An analysis of the effects of disturbances on the index illustrates how they may reduce profitability by restricting the ability of an MPC to reach dynamic equilibrium near process constraints, which in turn increases product quality giveaway and costs. Hence the index monetises the required control effort. Since linear models were used to predict the dynamic behaviour of chemical processes, which often exhibit significant nonlinearity, this Thesis also includes a new multi-model prediction method. This new method, called Simultaneous Multi-Linear Prediction (SMLP), presents a more accurate output prediction than the use of single linear models, keeping at the same time much of their numerical advantages and their relative ease of obtainment. Comparing the SMLP to existing multi-model approaches, the main novelty is that it is built by defining and updating multiple states simultaneously, thus eliminating the need for partitioning the state-input space into regions and associating with each region a different state update equation. Each state’s contribution to the overall output is obtained according to the relative distance between their identification point, i.e., the set of operating conditions at which an approximation of the nonlinear model is obtained, and the current operating point, in addition to a set of parameters obtained through regression analysis. Additionally, the SMLP is built upon data obtained from step response models that can be obtained by commercial, black-box dynamic simulators. These state-of-the-art simulators are the industry’s standard for designing large-scale plants, the focus of this Thesis. Building an SMLP system yields an approximation of the nonlinear model, whose full set of equations is not of the user’s knowledge. The resulting system can be used for predictive control schemes or integrated process design and control. Applying the SMLP to optimisation problems with linear restrictions results in convex problems that are easy to solve. The issue of model uncertainty was also addressed for the EMOP index and SMLP systems. Due to the impact of uncertainty, the index may be defined as a numeric interval instead of a single number, within which the true value lies. A case of study consisting of four alternative designs for a realistically sized crude oil atmospheric distillation plant is provided in order to demonstrate the joint use and applicability of both the EMOP index and the SMLP. In addition, a comparison between the EMOP index and a competing methodology is presented that is based on a case study consisting of the activated sludge process of a wastewater treatment plant